重难点16 数列的综合应用【十二大题型】
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".
(1)
"{}#pq,ª
(2)Ü
"{}Ŋŋ
1 + 2
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N
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ą, 1-1ń2023·Ōō·¬ŎŏŐňĀe
"{}#r n qsçN1 + 2 +34 = 25Nő3 +2N
4N5
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ą, 1-2ń2024·ÒŒœŔ·ï¬ň已知数列{}和{},其中 = 2, N,数列{ + }的前项和
为.
(1)若 = ,求;
(2)若{}是各项为正的等比数列, =
,求数列{}和{}的通项公式.
ą, 1-3ń2024·ŕŖ·ï¬ňÎ{}I
"NľrqsN{}I!
"Nő1 = 1 = 3N4 =
2N3 = 15.
(1){}5{}#pq,ª
(2)Î =
为奇数
·+1 为偶数
N
"{}#r qs ª
(3)ÜXŗ# N*+,( + 1)
(
)( + 2)
< 0-./N(
#Řëùk.
2
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6,9,11,13#ſö
¼]U´TƀƁĉçŵŶƂ NÝūŵŶƂ çń ň
A1157
B1177
C1155
D1122
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ƃŧŨƄƅũŪƆƇ
“ƈƉƊĊƋNƌēƍžZ
ńÇƎň”ę“ƈ” “Ɖ”5“Ǝ”Z#ç2 + 2 = 2ńľ' " ňį #N0ßƏƈ
ƉƎ
òƁ"
(3,4,5),(5,12,13),(7,24,25)N!NÝÏÊƁ"'Ní 10 ÊƈƉƎ
ò'#“Ǝ”
Xń ň
A145
B181
C221
D265
ą, 2-2ń2024·úŅ·¬ŎŏŐň·P>?IƐƑō
ŲƒƓƔ・ƕƖƗƘƙ¹Ï 20 ƚƛ 70 _
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A12
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D121
ą, 2-3ń2024·řŚƥ'·ï¬ňĂ 1 IíƦƧō)
ƨƩT§ńĄşICME
$ň#§ƪĂƫN§ƪ#
ÿĂƫIđßĂ 2 Øơ#[ƬƭƮƯøƯPưē.#Nľ'+,1 = ,1,2 = ,2,3 = = ,7,8 = 1NßƱ
ƲĂ 2 '#ƮƯøƯPƳƴĉƏtNÝíÊøƯP#=Ƶçń ň
A
2
B
2
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D
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A =
B = (1 + 5‰)12
C = (1 + 5‰)
D < < (1 + 5‰)12
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Ń 4ń2024·ǘŤIJȧ·¬ŎŏŐňĀe
"{}Ŋŋ1 +
2
2 +
3
3 + +
= ( 5).
(1)
"{}#pq,ª
(2)Āe
"{}Ŋŋ =
2+1.
%
"{}#r n qsª
&Ü+,(
) < +
2ŗ #-./N(
#Řëùk.
ą, 4-1ń2024·ǻǼǻǚ·¬ŎŏŐňĀe
"{}#rqsçN
"
Iç
1
2#
"Nő
1 = 2.
(1)
"{}#pq,ª
(2)ÜǪÏ #*NÆÇ
1
12 +
1
23 + +
1
+1 6 +1./N(
#Řëùk.
ą, 4-2ń23-24 ïƏ·ǘť·ķ'ňĀe
"{}#rqsçNőŊŋ = 2
.
"{}#rq
sçNőŊŋ1 = 1N
1
12 +
1
23 + +
1
+1 =
1
+1( N*).
(1)
"{},{}#pq,ª
(2)Ü = NÎ
"{}#rqsç7Nőŗ# N*,7
[
(
)4]+1 < 0-./N4#Ř
ëùk.
ą, 4-3ń2024·źȨ·¬ŎŏŐňĀe1 =
3
2N+1 =
(
)(
)(1
)( N)
(1)23
į 6 2N+1 = 2
+4ª
(2)ȩ =
N
ńiň23
į 6 2N
1
=
:=1
1
:ª
ńiiňIȤǪÏċ(
4NÆÇ|
1
| " 4-./NÜǪÏN4#êUëNÜ+ǪÏNșȦ3đ
5
"'#+,23&
Ń 5ń2024·Ōō·¬ŎŏŐňĀe
"{}#r n qs = 2
(1){}#pq,ª
(2)23
121
2 +
221
4 +
321
6 + +
21
<
5
4
ą, 5-1ń2024·ƶťȪȫȬ·ï¬ňĀe!
"{}#rqsçNő
"{ + 2}I!ç 2 #
!
".
(1){}#pq,ª
(2)Ü =
22
(21)+1N
"{}#r n qsçN2
<
1
2.
ą, 5-2ń2024·Ōō·¬ŎŏŐňĀe
"{}Ŋŋ3
1 + 3
2 + ... +3
+ = 4N N
(1)
"{}#pq,ª
(2)Ü =
N23
1
1 +
1
2 + ... +
1
<
7
9
ą, 5-3ń2024·ljǼ·ï¬ňƆç
"{}#rqsN2 =
1
4, +
1
2 = cos?
(1)3s{}#pq,ª
(2)Î
"
1
|| #rqsçN23
1
8
1
8 ×
1
4 <
A=1
1
A <
1
6
6 4
"&
Ń 6ń2024·ƶŤȭȮ·¬ŎŏŐňį:1,:2,,:A #*Nő:1 < :2 < < :ANţǨƲ:1,:2,,:A
ȯȰ
"{}#4
"Āe{}çċq!
"Nőľ!çB(B C 1)
(1)ƮȱÉw:A5A#TU
(2)IȤǪÏS#:1,:2,:3Ŋŋ
:1,:2,:3.!
"Nő4
":1,:2,:3ȗ.!
"DzÜǪÏNșȚw[ò
:1,:2,:3#ëªȤÝNșȦ3đ
(3)Ü = :1 + :2 + + :A(:A " 44 #*)N23
į1 = 1NB 6 2N02A
" < 4+1
ą, 6-1ń2024·ťȲŚNJ·ï¬ňĀe
",D1,2N],'Ří:E1q í:E2q … í:EAq:F1 < :F2
< < :FAA.
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:H = 1NÝşGĔ0noIþC
,#ŗ[qǔI,#F1Fq4"NőĔ0n
oI
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Ń 7ń2024·ȻŒ·¬ŎŏŐňĀeCDÏ R Ò#6
M(N)ŊŋM(N + O) = M(N)M(O)
M(N)
M(O) +6N
M(1) = 4NÝM(1) +M(2) + +M(99) = ń ň
A299 +198
B299 +196
C2100 +198
D2100 +196
ą, 7-1ń2024·ȼȽ·ï¬ňÎ
"{}#r n qsçNK N)Ï6
MN = ,N2
+GN + P,GP Q#ĂăÒNÝń ň
AP0 = 1
BÜ, = 0NÝR0 NNÆêT
CÜ, > 0NÝR0 NNÆêT
DÜ, < 0NÝR0 NNÆêT
ą, 7-2ń2024·ÒŒ·¬ŎŏŐňĀeM(N) =
1
2N2 +
1
2NN
"{}#rqsçNK()( N*)VÏ
6
O = M(N)#ĂăÒ.
(1)
"{}#pq,ª
(2)ÜT(N) =
4N
4N22Nȩ = T
2025 ( N*)N
"{}#r 2024 qs2024.
ą, 7-3ń2024·ǻǼ·[¬ňĀe
"{}#r n qsçNn çċČ
Nő3(
) = 4(
)
(1)2
"{
}I!
"Nƌ
"{}#pq,ª
(2)ÜKI
22
3
Ï6
O = log4N#ĂăÒNő
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+1
+1N
"{}#r n qs
8
"57
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Ń 8ń2024·Ōō·¬ŎŏŐňÎČ
W > 1NN >
őN C 0N6
MN = (1 + NW
WN
(1)23
MN > 0ª
(2)ÎN > 0N23
ln(1 + N < Nª
(3)Î #N23
1 + 2
1
2 + 3
1
3 + +
1
<
ln + 1)
ą, 8-1ń2024·ǻǼǼȾ·ø¬ňĀe
4 QNÎM(N) = lnN +
4
NN
(1)Ü4 = 1N6
O = M(N)Ï(1,1)ij#ȿɀª
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2
>
M(N1)
M(N2)
N1
N2
ą, 8-2ń2024·ljŚ·[¬ňĀe > 0Nő C 1N6
M(N) = N + ln(1 + N)
.
(1)Ɔ = M()
ln( + 1) +ç
"{}#rqs.23
į =
8
9N64 < 2024ª
(2)Ü =
1
eN23
NM(N) 6 0ª
(3)ÜM(N)0 3 ÊǎKN(
#Řëùk.
ą, 8-3ń2024·śƪɁɂ·¬ŎŏŐňĀe6
M(N) = ln(N + 1)T(N) =
N
N2Nľ' 6 1
(1)Ü = 1N23
N > 0NM(N) <
N
T(N21)ª
(2)Ü6
\(N) = M(N)
T(N)ÏľCDƟLŻż°ļN(
#ëª
(3)Āe
"{}#pq,ç =
]e
N2
> +1 > e
3
4
9
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Ń 9ń2024·ƣɃɄ·ï¬ňĐɅòɆefɇɈNĀeɉĆhí[#89ç
1
11N]íïƍɊNÜ
ɉĆr[hõNÝāh#89ç
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4ªÜr[hNÝāh#89ç
1
3.ƆɉĆɋhíh
õ#89çINį 6 2NI " ^-./NÝ^#êUëçń ň
A
97
132
B
49
132
C
47
66
D
49
66
ą, 9-1ń2024·ljǼɌɍ·[¬ňÜ
"{}#pq,ç =
NƆÏ
"{}#r + 2( N)
q'ŗŘ
ǔIċ
#89çINÝń ň
AI1 =
2
3
BI9 < I10
CI10 < I11
DI11 < I12
ą, 9-2ń2024·ǻǼǻǚ·¬ŎŏŐňɉ Ɏ ɏøɐ¡¢ƝɑɒɓNdžŮɔɕ[ɖoġVɗ#ċĕ
ɘ4²CƝɑ#,
įɑÏɉə'NÜɘ4K
TX 3NÝɉɑƝÉɎNÜK
+TX 3NÝɉɑ
ȠɚƳƴɔɕɘ4ªįɑÏɎə'NÜɘ4K
TX 4NÝɎɑƝÉɉNÜK
+TX 4NÝɎɑƝ
ÉɏªįɑÏɏə'NÜɘ4K
TX 3NÝɏɑƝÉɉNÜɘ4K
+TX 3NÝɏɑƝÉɎ.ǮɊ
NɑÏɉə'.
(1)øŮɔɕɘ4ĥɑÏɉə'#89ª
(2)ɔɕ( #*)Ůɘ4ĥNƆɑÏɎə'#89çWN
"{W}#pq,ª
(3)Î =
1
(
)WW+1N2
1 + 2 + + "
4
3.
ą, 9-3ń2024·Ōō·¬ŎŏŐňɉ ɎƃUɛɜNdžɐə'Ɗ0 3 ɝɃ_ƛËɞɟNľ'ɉə'# 3
ɝɞɟç 1 ɝǟɠs 2 ɝɡɠNɎə'# 3 ɝɞɟǔIǟɠ#NxÏɐƊ]Ċĭ#ɞɟ'ɢǀŘ 1 ɝN
t5$ɣNźLjŮS#ɤĉNƆɉə'ɡɠƛËɞɟNɝNɥ0 2 ɝɡɠƛËɞɟ#89çWNɥ
0 1 ɝɡɠƛËɞɟ#89çB
(1)W2,B2#ë
(2)&ɤĉ>Ůɉə'ɡɠƛËɞɟɦåĈʼnŮwx 0 ɝNʼnŮwxRE`#89W
(3)Ɔ = 2W + B
ńiň23
"{
}ç!
"Nƌw{}#pq,
ńiiňN#·Ƙ"d
ķɧńȣơň
10
"5<=>?#$%&
Ń 10ń23-24 øƏ·Ōō·ȴȵȶȷňĀe!
"{}#!çBNrqsçN > 0N22 + 3 =
4N5 = 44
ń1ň
ń2ňÏ<=ƮƯɨǕN+O'NÎK_A(AA)A = 1,2,3 )NƮɀ_A_A+1#ɩ9ç2ANő1 = 1N
"{}
#pq,
ą, 10-1ń2024·úŅÛǚ·ï¬ňĀeɪɀ`:O2 = WNW > 0)NƮɀaDO = AN
W5`$X,GKN
ɀȵ AB 'K^(N4,O4)AO4 = 2.
(1)ɪɀ`#ª
(2)Ʈɀa5Nv$XKP+çďKNÎ b G+P b P+^ b ^+,#=Ƶ·ǩçbG+P,bP+^,b^+,NÜbG+P,
bP+^,b^+,.
"NA.
ą, 10-2ń2024·śƪHɫ·ï¬ňĀe{}IƊqVçċ
#!
"Nő1 + 2 = 3N3
2 = 2N
"{}#rqsçNő3 = 5N4 = 16.
ń1ň
"{}N{}#pq,ª
ń2ňßĂÏ<=ƮƯɨǕ'NKI1(1,0)NI2(2,0)N…I(,0)NI+1(+1,0)N
_1(1,1)N_2(2,2)N…N_(,)NÜƆ b I_I+1#=ƵçN
"{}#rqs.
ą, 10-3ń2024·úŅLɄ·¬ŎŏŐňĀeɬȄPDN2
2 + O2
2 = > > 0)NKIIɬȄÒ#ĦKN\1,\2I
ɭ ɮɯKNLI b I\1\2#ź¶NőL´K^
1
3 ,0 5K# 1
3 ,0 #ɰæZsç
4
3
(1)ɬȄ C #ª
(2)ÜƮɀ l ÈKI(4,0)N5ɬȄ$X ANB KÜ|,I|,|,G|,|GI|.!
"Ncosc,\2G#ë
11
"'#@A+B
Ń 11ń24-25 ïÒ·Ōō·ɱĥĉɲňĀe{}I
"Nľr n qsçN4 =
Nĩ]Äî%
4 =
ªÄî&
1 = 23.ÊÄî'Þ[ÊĉçĀeN
(1)
"{}#pq,ª
(2)#êUëNƌįŘÇêUë n #ë.
ą, 11-1ń2024·ȻŒŚȽ·ï¬ňĀe
"{}N_______________.ș]Ə"ÊÄî'ŗ[ÊNɳ
ɴÏÒ=#&'ƌ1h.ń
ßƱÞQÊÄîN¼¾í[Ê1hÉ·.ň%
"{}#rqsç = 2
ń #ňª&
"{}#rqZƵç, = 2
(+1)
2
ń #ň.
(1)
"{}#pq,ª
(2)ȩ = + log2N
"{}#rqs.
ą, 11-2ń2024·úŅƖŇ·ø¬ňĀe{}I
"N{}I!
"Nő{}#rqsçN21 =
1 = 2N5 = 5(4
3)NÏ%5 = 4(4
3)N&+1 = +2ÊÄî'ŗľ'[ÊNǯ.Ə=&
#1h.
(1)
"{}s{}#pq,ª
(2)Î
"
#rqsçN.
ą, 11-3ń23-24 ïƏ·ťȲɵɶ·ķſňĀe
"{}#rqsçNő4 = 10,3 = 18.
(1)
"{}#pq,ª
(2)ÜƊqVçċ
#
"{}ľrqsçNĩ]Äî% Äî& ÄîdøÊÄî'Þ[Êĉç
ĀeNÎ = + N
"{}#pq,s
"{}#rqs^.
Äî%
= 3
ª
Äî&
1 = 2,
+1
= 3ª
Äîd
J 6 2ő Zǔ02
=
+1./N1 = 2,3 = 3.
12
"#CD EF&
Ń 12ń2024·ƶťɝŲɷ·ï¬ňßƱq
Ć#
"{},{}Ŋŋ{} f {} = {
}Nő:çɸ
N: < :ª:çɹ
N: > :Nľ': {
}NǖǗɦş{},{}ç“ɳ$ɺ
"”NƆ 1,2
1,2
ç{},{}#“ɳ$ɺ
"”Nç{}#rqs.
(1)Ü{} f {} = {1,2,3,4,5,6}Nő1 = 5NȚwØ0ŊŋÄî#“ɳ$ɺ
""ª
(2)į{},{}ç“ɳ$ɺ
"”N
ńiň23
ŘêTëNǪÏ: = ª
ńiiňįçɹ
N#êTë.
ą, 12-1ń2024·ɻɄ·¬ŎŏŐňĀeċČ
4NÎ1N2N…N 4N1N2N…N 4I4Êɼɽ
(
N =
4
:=1: =
4
:=1: > 0.ÜXŗ: = 1,2, 4NŘ 4+1 = 1N 4+2 = 2N 4+1 = 1Nǔ
0::+2 6 : + :+1NÝş4Ê
A.(4)—ɾĢ
ò.
(1)Țw 8 Ê+Ō#
NÆÇ 8 Ê
A.(8,2)—ɾĢ
òª
(2)êU#NÆÇ1N2N…N6N1N2N…N6A.(
)—ɾĢ
òª
(3)Ü4 6 4Nő1N2N…N 4N1N2N…N 4A.( 4)—ɾĢ
òN:(: = 1,2, 4)#êT
ë.
ä,
ńiň(N1 + N2 + N3)2 6 3(N1N2 + N2N3 + N3N1)NįőɿįN1 = N2 = N3Řªńiiňįċɹ
6 4
NÎ = A(A #*)N0N1N2 + N2N3 + + NN1 " (N1 + N3 + + N A
)(N2 + N4 + + N A)ªįċɸ
> 4
NÎ = A + 1(A #*)N0N1N2 + N2N3 + + NN1 " (N1 + N3 + + N A+1) (N2 + N4 + + N A).
ą, 12-2ń2024·śƪʀǘ·¬ŎŏŐňX
"{},{}NßƱǪÏċČ
0 6 3NįŗċČ
" 0
V01 < 1 < 2 < 2 < … <
< < NÝş{}ç{}#“0q°ļʁ
"”Ü0åŘŗ#ċ
Č
NÝş{}ç{}#“ʂȸ°ļʁ
"”
(1)Āe = 2NșȚw[Ê
"{}#“ʂȸ°ļʁ
"{}”NƌȦ3đDz
(2)Ü{},{}Ŋŋ + =
Nľ'{}Iʼnq1 = 1#
"Nį{}ç{}#“ʂȸ°ļʁ
"”
N{}#pq,
(3)Āe
"{}sċČ
!
"{}Ŋŋ
= A
A + 1)
= 1,2,…,2024)Nľ' k IċČ
N2
ǪÏċČ
kNÆÇ{}ç{}#“2024 q°ļʁ
"”
ą, 12-3ń2024·ʃ·ï¬ňţǨƲŊŋƏ"Äî#
"{}şç4
h
"
%
"{}#dž[qǔIċɹ
ª
&ǪÏċɸ
mNÆÇ
"{}#dž[qžc m ØÇ#ȭǔ+Iċɹ
(1)Ü aNbNc Iç 2 #
"N2
aNbNc +I
h
"ª
(2)Ü
"{}ŊŋŗċČ
pNqN-0W+B =
1
W + 1
B WBNő1 = 8NȺĨ
"
IȤI$
h
"N
ƌ23ȟ#@Ъ
(3)ĀeƊqVçċ
#
"{}Ÿ0 100 qNőŗ1 " " 100N-01 + 2 + + =
3
123
2223
A42A3
12A3
222A3
2A2(A N*)NÜ
"{}ç
h
"NŊŋÄî#Ø0ö
k ë#s
[ Ż
1ń2024·LʄŬʅʆ·ø¬ňÎç
"{}#r n qsNÜ5 = 41N1 > 0NÝÆ > # n #ê
Tëçń ň
A11
B12
C20
D21
2ń2024·ljǼɌɍ·ï¬ňĀe{}I
"N1 = 3,4 = 12NÏ
"{}'1 = 4,4 = 20NÜ{
}
I!
"NÝ2024#ëçń ň
A6072
B22023
C22023 +6072
D22023
$
3ń2024·úŅ·¬ŎŏŐňŤʇ
Ųʈʉ#ź}ŧĉŨʊ1ʋ¿Ū'#“ʌƵȳ”&ʍʎěȴ
"cȴ
"'#ïȴ
"çŃNľ¹KI]
"'#íïqƍɊNdž[q5r[q#
A.
"ÜĐÊïȴ
"#r 4 qç1,4,8,13NÝū
"#í 18 qçń ň
A188
B208
C229
D251
4ń2024·ȻŒŚȽ·[¬ň
"{}'#2N2024I6
M(N) = N3
N2 +N
#ʏëKNÝlog21013
= ń ň
A
1
2
B1
C
D
1
2
5ń2024·Ōō·ï¬ň
"{}#ɸ
q.!
"Nɹ
q.!
"NI
"{}#rqsN1
= 3N2 = 2N3 = 4N4 = 7NÝń ň
A A < A+1A N*NőA 6 2
Bį 6 5Nő N*N
"{}I°÷
"
C
10
11 < 10
D100 < 9
6ń2024·Ōō·¬ŎŏŐňĀe #N =
1
N =
1
21)2
N
"{}5
"{}#Ÿq¼]T´
U#ǞƁƀ"ò.[Ê
"{}NÝ
"{}#r 99 qsçń ň
A
196
197
B
198
199
C
98
197
D
99
199
7ń2024·ǘť·ï¬ňĀe
"{}#r n qsçNő = 2 +4N N*NÜXŗ# [0,1]N
+,
< N2
(1 + )N
2
+ 2-./NÝ(
x åĈçń ň
A
B0
C1
D2
8ń2024·LʄŬʐʑ·ø¬ňßĂIʒʓ
ŲʔÏ 1904 _A«#ĈʕʖʗʘȅPʙ#Ăƫ.ĂP#ĉ
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ĉċøƯPNĩtʝȑʚ.ûLj¡¢[ÈNɦÇ´[Ä“ʘȅ”ʙ#ʞɀNşçʔʞɀ.Î Pn #ƄIJs=
Ƶ·ǩç Ln SnNƏ"@Ğċħ#Iń ň
%P₅#ʚ
ç3 × 44;
&h5 = 3 ×
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3
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;
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jR# > 0, < #
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ï Q
9ń2024·ȼȽʠŇ·¬ŎŏŐňĐɐƽ[ʡ 15 DŽDž#ʢNƼƽįŕȒĸ 3 DŽDžʼnĸNŠšʜɡ¢ʣĹN
LJ90.3%N· 12 ÊLJʤ»ńdžLJƼƽʢ#ǖ[ŕ·ķʤĹň.0RǟǠƫ
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<V·ʥ´dž[ķ¡¢ʦʤNdž[ķØʤĹǟǥđǸ·ò.N[Ǹ·çdžķǟNęʣĹǟžcʤĹ
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Nʧ[Ǹ·I×NęʣĹǟ5ĀʤǟȎǥ#Ƌc9ªǥ×ʤĹNdž[ķʦʤĆ
ǥ
#×sN×cLj;.Ə"Ȧċħ#Iń ň
AǥǟƫNØ0#×sç 2340 Dž
BǥǟƫNêĥ[ÊLJʤĹǟǥç 10030 Dž
Cǥ×ƫNdžLJʤĹǟǥ'#ǟǸ·ʨx°ļ!
"
Dǥǟƫ!ǥ×ƫʤĹ×ĴʩNØcǥǟƫĮXǥ×ƫ
10ń2024·ʪʫ·¬ŎŏŐňĀeÏ+ç 0 #
"{}'N4 =
5I256#!'qN
"{}
#rqsçNő =
1
+1NÝń ň
A =
BJ N*, 6
C =
1
11
1
DJ N*,6 " " 5
11ń2024·ɻɄȽʬ·¬ŎŏŐňĀe
"{k}Nľr n qsçNÜǪÏ
^ > 0Nŗ# NN-
0|k+1
k| + |k
k
| + + |k2
k1| " ^NÝş{k}çG
".ÝƏ"Ȧċħ#Iń ň
AÜ{k}Ic 1 çʼnqNB;B;<1)ç!#!
"NÝ{k}çG
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BÜ{k}çG
"NÝ{}ȗçG
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CÜ{}çG
"NÝ{k}ȗçG
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DÜ{},{}VçG
"NÝ{ }ȗçG
"
ø ʭʮ
12ń2024·ƶť·¬ŎŏŐňĀe
"{}Içl#
"NƊq+#
"{}Iʼnqç 2N!ç
B#!
"Nő2 = 1,3 = 2NÝl B =
13ń2024·ƶŤƷƸƹ·ï¬ňCD
X6
M(N)s
"{N}NÜ(N+1
N)MZ(N) +M(N) = 0NÝş
"
{N}Ĕ0“M(N)6
no”.ĀeïŮ6
M(N)Ăă#êʯKç(
)NőM(N + 1) = M(N) + N + 1NÜ
"{N}
Ĕ0“M(N)6
no”Nőʼnqç 1 #
"{}Ŋŋ = ln(N + 2)
ln(N
)NƆ{}#rqsçNÝ
"
2
#êUëç
.
14ń2024·Ʉʰʱǚ·¬ŎŏŐňX0ʲ
"{}N]
"{}'Ří:1q、í:2q、、í:4q
(:1 < :2 < < :4)NǞŮƀ"A.
"{A}Nľ'A = :A,1 " A " 4NÝş
"{A}ç{}#[Ê4"Nş{A}
ƊqZsç{}#[Ê4"sþC
"{}#ŗ[qǔI{}#4"Ý
"1,2,4,8,16,32#Ø04"
s#sç
ú 1h
15ń2024·ǘŤ·¬ŎŏŐňĀe
"{}#rqsç,3+1 = +1,2 =
1
9Nċq
"{}Ŋŋ1
= 2Nő1,2
3.!
".
(1){}s{}#pq,ª
(2)23
1 + 2 + + <
3
26.
16ń2024·Ʉʰʂʳ·ï¬ňĀeċq
"{}#rqsçNŊŋ2 = 2
+
.
(1)
"{}#pq,ª
(2)Î =
2,ç
"{}#rqs.Ü
3
2
"
A(23)
ŗ# #*-./N k #Řëùk.
17ń2024·úŅʴlj·ø¬ňßĂNK,:(: # )VÏ x v#ċʵvÒN b +,1G1N b ,1,2G2…N b ,
G
P·ǩIc1N2N…Nń # ňçʚIJ#ʚøƯPNőʶKG:(K # )VÏ6
O =
N#ĂăÒ
(1)1N2N3#ëNƌȚw{}#pq,ń+23ňª
(2)
"
1
+1 #r n qs
18ń2024·ǻŚŤȽ·ø¬ňʷʸŕʹʺJNɅçøǑ#ĆŀǑěʻʼʽsɡʾʿRʴˀNĐĆ
džŕǔ§ÏRʴˀ'Þ[RNĀeūĆí 1 ŕÞʻʼʽ#89I
2
3NÜr[ŕÞʻʼʽNĥ[
ŕƳƴÞʻʼʽ#89ç
1
3Nēr[ŕÞɡʾʿNĥ[ŕƳƴÞɡʾʿ#89ç
1
2NßāȘLj
(1)ūĆí 2 ŕÞʻʼʽ#89ª
(2)ƆūĆíŕÞʻʼʽ#89çIN23
I
3
7 ç!
"ª
(3)]í 1 ŕ´í 10 ŕ'NūĆÞʻʼʽ#89TXÞɡʾʿ89#ŕ
19ń2024·ƶŤ·ø¬ňĀe
"{}#rqsçNÜǪÏ
> 0)NÆÇ 6 +1ŗ #
ǔ./NÝş
"{}Ĕ0noI
(1)Ü
"{}ç
"Nő3 =
!5 =
N2
"{}Ĕ0noI
ª
(2)Î
"{}#ƊqVçċ
Nő{}Ĕ0noI
%Ü
"{}I!çB#!
"Nő = 4NB#ëª
&#êUë